When does elementary bi-embeddability imply isomorphism?

2007/05/13 by Goodrick, John
#03C45 (Primary) 03C52 (Secondary) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.0705.1849

Abstract

A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if a countable theory T has the Schroder-Bernstein property then it is classifiable (it is superstable and has NDOP and NOTOP) and satisfies a slightly stronger condition than nonmultidimensionality, namely: there cannot be a model M of T, a type p over M, and an automorphism f of M such that for every two distinct natural numbers i and j, fⁱ(p) is orthogonal to f^j(p). We also make some conjectures about how the class of theories with the Schroder-Bernstein property can be characterized.

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